Modeling an Asset Price
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MODELING AN ASSET PRICE
By Adnan Kuhait MSc Business Analytics
Course: Applied Analysis : Financial Mathematics Vrije Universiteit (VU) Amsterdam
Reference book: The mathematics of Financial derivatives, A student introduction by Paul Wilmott, Sam Howison and Jeff Dewynne Cambridge University Press
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The absolute change in the asset price is not by itself useful
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The absolute change in the asset price is not by itself useful
for example a change of 1$ is more significant in an asset price of 20 $ than if it is 200 $
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Return
so we define a “return” to be the change in the price divided by the original price
this relative measurement is better than the absolute measurement
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Return
so we define a “return” to be the change in the price divided by the original price
this relative measurement is better than the absolute measurement
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in a small interval of time dt, asset price changed to S+ds
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the return will be ds/S this has two parts:
First part predictable, deterministic and anticipated return just like investing in risk-free bank which equal to: ds/S = µ dt where µ is the average rate of growth of the asset price (called the drift) and it is considered to be constant. (in exchange rated it could be a function in S and t.
ds / S = µ dt
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the return will be ds/S this has two parts:
Second part is the random change in the asset price in response to external effects. it is represented by a random sample drawn from a normal distribution with mean zero. (σ dX) where σ sigma is the volatility, measure the standard deviation of the returns. dX is the sample from the normal distribution.
ds / S = µ dt + σ dX
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the return will be ds/S this has two parts:
Second part is the random change in the asset price in response to external effects. it is represented by a random sample drawn from a normal distribution with mean zero. (σ dX) where σ sigma is the volatility, measure the standard deviation of the returns. dX is the sample from the normal distribution.
Or ds = µ S dt + σ S dX
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by taking sigma = 0 we will be left with ds= µ dt or ds/dt= µS since µ is constant, then this can be solved: S= S0e(t-t0) where S0 is the asset price at t0
the asset price is deterministic and can predict the future in certainty.
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Wiener Process
dX , the randomness, is called Wiener Process.
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Wiener Process
dX , the randomness, is called Wiener Process.
Wiener Process properties: • dX is a random variable, drawn from a normal
distribution. • the mean = 0 • the variance is = dt
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Wiener Process
dX , the randomness, is called Wiener Process.
Wiener Process properties: • dX is a random variable, drawn from a normal
distribution. • the mean = 0 • the variance is = dt It can be written as:
𝑑𝑋 = 𝜑 𝑑𝑡 Where φ is a random variable drawn from standardised normal distribution which has zero mean and unit variance and pdf:
1
2𝜋 𝑒
−12 𝜑2
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Wiener Process
dX , the randomness, is called Wiener Process.
Wiener Process properties: dX is scaled by dt because any other will lead to either meaningless Or trivial when dt0. which we are in particularly interested and it fits the real time data well
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suppose todays date is t0and asset price is S0. if in the future time t1and asset price S1, then S1will be distributed around S0 in a bell shaped graph. the future price will be close to S0 the further t1 is from t0the more spread out this distribution is. if S represents the random-walk given by ds= µ S dt + σ S dX then the probability density function represented by this skewed curve is the lognormal distribution, and therefore ds= µ S dt + σ S dX is the lognormal random-walk. properties of the model: does not refer to the past history of the asset price, next asset price depends only on today’s price.(Markov properties)
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in real life prices quoted in discrete time intervals, but for efficient solution we use continuous time limit dt 0 Result :
𝑑𝑋2 → 𝑑𝑡 𝑎𝑠 𝑑𝑡 → 0 𝑤𝑖𝑡ℎ 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 = 1
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Ito’s Lemma
suppose f(S) is a smooth function of S using Taylor’s series:
𝑑𝑓 = 𝑑𝑓
𝑑𝑆 𝑑𝑆 +
1
2 𝑑2𝑓
𝑑𝑆2 𝑑𝑆2 + …
But dS = µ S dt + σ S dX Then (𝑑𝑆)2 = (µ S dt + σ S dX)2
(𝑑𝑆)2 = µ2 𝑆2𝑑𝑡2 + 2σµ𝑆2dtdX + 𝜎2𝑆2𝑑𝑋2
Since 𝑑𝑋 = 𝑂 𝑑𝑡
Then the last term is largest for small dt and dominate the other two terms. So we have: 𝑑𝑆2 = 𝜎2𝑆2𝑑𝑋2 Using the result 𝑑𝑋2 → 𝑑𝑡 then:
𝑑𝑆2 → 𝜎2𝑆2 𝑑𝑡
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Ito’s Lemma Substituting in Taylors expansion:
𝑑𝑓 = 𝑑𝑓
𝑑𝑆 𝑑𝑆 +
1
2 𝑑2𝑓
𝑑𝑆2 𝑑𝑆2
𝑑𝑓 = 𝑑𝑓
𝑑𝑆 µ S dt + σ S dX +
1
2 𝑑2𝑓
𝑑𝑆2 (𝜎2𝑆2 𝑑𝑡)
Or
𝑑𝑓 = σ S 𝑑𝑓
𝑑𝑆dX + (µS
𝑑𝑓
𝑑𝑆+
1
2𝜎2𝑆2
𝑑2𝑓
𝑑𝑆2)𝑑𝑡
relating the small change in a function of random variable to the small change in the variable itself
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Ito’s Lemma
this can be generalized to f(S,t)
𝑑𝑓 = 𝑑𝑓
𝑑𝑆 𝑑𝑆 +
𝑑𝑓
𝑑𝑡 𝑑𝑡 +
1
2 𝑑2𝑓
𝑑𝑆2 𝑑𝑆2
And doing the same, we will get:
𝒅𝒇 = σ S 𝒅𝒇
𝒅𝑺dX + (µS
𝒅𝒇
𝒅𝑺+
𝟏
𝟐𝝈𝟐𝑺𝟐
𝒅𝟐𝒇
𝒅𝑺𝟐+
𝒅𝒇
𝒅𝒕)𝒅𝒕
Which is Ito’s Lemma
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Black-Scholes equation
suppose we have an option V(S,t) depends on S and t using Ito’s Lemma:
𝑑𝑉 = σ S 𝑑𝑉
𝑑𝑆dX + (µS
𝑑𝑉
𝑑𝑆+
1
2𝜎2𝑆2
𝑑2𝑉
𝑑𝑆2+
𝑑𝑉
𝑑𝑡)𝑑𝑡
assuming V has at least one t derivative and two for S
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Black-Scholes equation
consider the portfolio: ∏ = V - ∆S then one jump in the value in one time-step is: d∏ = dV - ∆dS Where ∆ fixed in the interval [t,t+dt]
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Black-Scholes equation
using Ito’s Lemma for the value of dV and the first model for dS: d∏ = dV - ∆dS
𝑑∏ = σ S 𝑑𝑉
𝑑𝑆dX + µS
𝑑𝑉
𝑑𝑆+
1
2𝜎2𝑆2
𝑑2𝑉
𝑑𝑆2+
𝑑𝑉
𝑑𝑡𝑑𝑡
− ∆ (σ S dX + µ S dt )
= 𝜎𝑆𝑑𝑉
𝑑𝑆 − ∆ dX +(µS (
𝑑𝑉
𝑑𝑆− ∆) +
1
2𝜎2𝑆2
𝑑2𝑉
𝑑𝑆2+
𝑑𝑉
𝑑𝑡)𝑑𝑡
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Black-Scholes equation we want ∏ to be a bond thus it should be deterministic, that's mean the random term dX should be dropped. so take
𝑑𝑉
𝑑𝑆 = ∆
Then:
𝑑∏ = (1
2𝜎2𝑆2
𝑑2𝑉
𝑑𝑆2+
𝑑𝑉
𝑑𝑡)𝑑𝑡
But:
∏= ∏0ert
then d∏= r∏0ertdt d∏= r∏dt So
(1
2𝜎2𝑆2 𝑑2𝑉
𝑑𝑆2 +𝑑𝑉
𝑑𝑡)𝑑𝑡= r∏dt
(1
2𝜎2𝑆2 𝑑2𝑉
𝑑𝑆2 +𝑑𝑉
𝑑𝑡)𝑑𝑡= r(V-∆S)dt
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Black-Scholes equation (
1
2𝜎2𝑆2 𝑑2𝑉
𝑑𝑆2 +𝑑𝑉
𝑑𝑡)𝑑𝑡= r(V-∆S)dt
But 𝑑𝑉
𝑑𝑆 = ∆
(1
2𝜎2𝑆2 𝑑2𝑉
𝑑𝑆2 +𝑑𝑉
𝑑𝑡)𝑑𝑡= r(V-
𝑑𝑉
𝑑𝑆 S)dt
Dividing by dt 1
2𝜎2𝑆2 𝑑2𝑉
𝑑𝑆2 +𝑑𝑉
𝑑𝑡= rV-
𝑑𝑉
𝑑𝑆𝑟 S
Or
𝒅𝑽
𝒅𝒕+
𝟏
𝟐𝝈𝟐𝑺𝟐 𝒅𝟐𝑽
𝒅𝑺𝟐 + 𝒅𝑽
𝒅𝑺 𝒓𝑺 − 𝒓𝑽 = 𝟎
Which is the Black-Scholes equation